Determinants

Determinants: 2×2 Determinants

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Determinants are like matrices, but done up in absolute-value bars instead of square brackets. There is a lot that you can do with (and learn from) determinants, but you'll need to wait for an advanced course to learn about them. In this lesson, I'll just show you how to compute 2×2 and 3×3 determinants. (It is possible to compute larger determinants, but the process is much more complicated.)
If you have a square matrix, its determinant is written by taking the same grid of numbers and putting them inside absolute-value bars instead of square brackets:

If this is "the matrix A" (or "A")...	...then this is "the determinant of A" (or "det A").

Just as absolute values can be evaluated and simplified to get a single number, so can determinants. The process for evaluating determinants is pretty messy, so let's start simple, with the 2×2 case.

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For a 2×2 matrix, its determinant is found by subtracting the products of its diagonals, which is a fancy way of saying in words what the following says in pictures:

the matrix A	the determinant of A ("det A")
the matrix A	the determinant of A ("det A")

In other words, to take the determinant of a 2×2 matrix, you multiply the top-left-to-bottom-right diagonal, and from this you subtract the product of bottom-left-to-top-right diagonal.
"But wait!" I hear you cry; "Aren't absolute values always supposed to be positive? You show that second matrix above as having a negative determinant. What's up with that?" You make a good point. Determinants are similar to absolute values, and use the same notation, but they are not identical, and one of the differences is that determinants can indeed be negative.

• Evaluate the following determinant:
I multiply the diagonals, and subtract:   Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved

• Find the determinant of the following matrix:

I convert from a matrix to a determinant, multiply along the diagonals, subtract, and simplify:


Determinants: 3×3 Determinants (page 2 of 2)
Sections: 2×2 determinants, 3×3 determinants

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The computations for 3×3 determinants are messier than for 2×2's. Various methods can be used, but the simplest is probably the following:   Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved

Take a matrix A:

Write down its determinant:

Extend the determinant's grid by rewriting the first two columns of numbers:

Then multiply along the down-diagonals:

...and along the up-diagonals

Add the down-diagonals and subtract the up-diagonals:

And simplify:

Then det(A) = 1.

• Find the deteriminant of the following matrix:

  

First I convert from the matrix to its determinant, with the extra columns:

Then I multiply down and up the diagonals:

Then I add the down-diagonals, subtract the up-diagonals, and simplify for the final answer:

There are other methods for simplifying determinants by hand, and these other methods are required when evaluating larger determinants by hand, but those methods can probably wait until later. For the time being, note that your graphing calculator should be able to evaluate the determinant of any (square) matrix you enter. For instance:

But make sure, even if you have a graphing calculator, that you can evaluate 2×2 and 3×3 determinants, because you are likely to have word problems where the determinants contain variables that your calculator can't handle.